Mr Burris will swear under oath that he did not promise anything to get the job
Roland Burris has said he expects to take up Barack Obama's vacant Illinois senate seat "very shortly".
Senate leaders had previously tried to block Mr Burris's membership, concerned that he had been picked by scandal-hit Illinois Governor Rod Blagojevich.
But after a meeting on Wednesday with Senate leaders, Mr Burris expressed confidence he would be allowed to take his seat.
Mr Obama said he would be happy to work with Mr Burris if he was seated.
On Tuesday, when the new Senate met for the first time, Mr Burris was barred entry to the chamber by officials, who said he did not have the correct credentials.
Illinois's secretary of state has so far refused to co-sign the letter of appointment from Mr Blagojevich, as required by Senate rules.
But following his meeting with Senate Majority Leader Harry Reid on Wednesday, Mr Burris said he was "very happy" and expected to come to an agreement with Senate Democrats.
After the meeting, Mr Reid laid out the legal steps needed for Mr Burris to take his seat.
If Mr Burris swears under oath at a committee hearing on Thursday that he did not offer Mr Blagojevich anything in return for the job, and if an Illinois court rules that Mr Burris does not need the state's Secretary of State to certify his appointment, then the Senate could review its position, Mr Reid told reporters.
Mr Obama has attempted to keep out of the row, but said he would be prepared to co-operate with Mr Burris.
"He's a fine public servant," he said at a Washington news conference.
"If he gets seated, then I'm going to work with Roland Burris just like I worked with all the other senators to make sure that the people of Illinois and the people of the country are served."
Mr Blagojevich is facing calls for his impeachment over allegations he tried to "sell" Mr Obama's vacant Senate seat. He denies any wrongdoing.